Wrapping my mind around vectors?

So, I'm assuming that scalars are what I'm used to working with in math. You add, subtract, multiply, etc.; they follow the rules I know. 1 + 1 = 2.

Scalar = Magnitude

Vector = Magnitude and Direction

Now, how do magnitude and direction coexist? Right now I'm just seeing some scalar volume cube being moved in a certain direction. What is the vector DOING? Can you provide me with a solid example of vectors at work?

A vector measures displacement... the distance from start to ending point. So how can they have arrows indicating their direction? They must be finite, but they look like rays.

It doesn't matter where you put a vector on cartesian or polar coordinates... but then we're supposed to calculate the vector using scalar components. If Vector C is 5 long, how can the square root of (really big number A squared) + (really big number B squared) equal really small number 5?

In addition, what ARE vectors? I mean, why are we taking the displacement instead of the distance, and adding some angle?

So, my main goal in asking these questions is to determine WHAT a vector is, why they exist, and how they differ from scalars.

What I have so far:

The purpose of vectors is to make it simpler to deal with "things" that have both a distance and a location... so velocity HAS to have a location? Wow, it really is not at all like speed.

OK, why do I want to know it's location?

Let's see... we put the vectors onto coordinate systems, like the cartesian coordinates. And we use scalars to plot things on the cartesian coordinates, so we break down the vector into components in order to plot it on the cartesian coordinate system.

Does this mean that the cartesian coordinate system is made up of scalars?

So, my main goal in asking these questions is to determine WHAT a vector is, why they exist, and how they differ from scalars.

What I have so far:

The purpose of vectors is to make it simpler to deal with "things" that have both a distance and a location... so velocity HAS to have a location? Wow, it really is not at all like speed.

OK, why do I want to know it's location?

Let's see... we put the vectors onto coordinate systems, like the cartesian coordinates. And we use scalars to plot things on the cartesian coordinates, so we break down the vector into components in order to plot it on the cartesian coordinate system.

Does this mean that the cartesian coordinate system is made up of scalars?

Velocity has a magnitude and direction. This is important because when two vector quantities interact with each other the direction they move is important. It is simpler than you are making it. Lets say that you have a plane flying at x mph and there is a wind directly behind it blowing at y mph. The wind will add to the planes speed. If the wind is blowing directly toward the plane it will subtract from its speed. If the wind is blowing at an angle toward the plane it will push the plane in the direction of the angle. The direction involved in vectors is just an easy way of taking all of this into account when you are solving this problem of how the wind and plane will interact.